Percolation on infinite percolation clusters

Let's consider an infinite percolation cluster $\mathcal{C}_p$ that takes shape in the supercritical phase ($p>p_c$) of a bond percolation in $\mathbb{Z}^d$. What could be said about a bond percolation on $\mathcal{C}_p$ parametrized by $p$ (or parametrized by a $q\ne p$)? Phase caracterizations? Critical values $q_c=f(p)$? Number of infinite clusters if any?

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Can't you just imagine running percolation on $\mathbb{Z}^d$ twice (independently), and losing edges that get deleted either time? So you're running percolation with parameter $p^2$ (or $pq$)? –  John Engbers Jun 1 '12 at 16:53
I think you're right John! Nothing interesting is emerging from this percolation :-) I should have thought more carefully before posting this question! –  user16782 Jun 1 '12 at 19:41
It turns out, however, that carrying out percolation on the percolation clusters at criticality leads to an interesting model! arxiv.org/abs/1410.3603 –  j.c. Oct 15 '14 at 1:01